Abstract differential geometry is a branch of mathematics that studies geometric structures on manifolds in a more general and abstract setting, primarily using concepts from differential geometry and algebraic topology. It emphasizes the intrinsic properties of geometric objects without necessarily attributing them to any specific coordinate system or representation. Some key features of abstract differential geometry include: 1. **Smooth Manifolds**: Abstract differential geometry focuses on smooth manifolds, which are spaces that locally resemble Euclidean space and possess a differentiable structure.

The Atiyah–Hitchin–Singer theorem is a result in the field of differential geometry and mathematical physics, particularly in the study of the geometry of four-manifolds. Specifically, it relates to the topology and geometry of Riemannian manifolds and their connections to gauge theory.

In the context of differential geometry and manifold theory, "density" generally refers to the concept of a volume density, which provides a way to measure the "size" or "volume" of subsets of the manifold. Specifically, there are several related ideas: 1. **Volume Forms**: On a smooth manifold \( M \), a volume form is a smooth, non-negative differential form of top degree (i.e.

The Chern–Simons form is a mathematical construct that arises in differential geometry and theoretical physics, particularly in the study of gauge theories and topology. It is named after the mathematicians Shiing-Shen Chern and James Simons. In essence, the Chern–Simons form is a differential form associated with a connection on a principal bundle, and it helps in the definition of topological invariants of manifolds, notably in the context of 3-manifolds.

In mathematics, particularly in differential geometry and the study of dynamical systems, the term "contact" often refers to a specific type of geometric structure known as a **contact structure**. A contact structure can be thought of as a way to define a certain kind of "hyperplane" or "half-space" at each point of a manifold, which has important implications in the study of differentiable manifolds and their properties.

The term "coordinate-induced basis" generally refers to a basis of a vector space that is derived from a specific coordinate system. In linear algebra, particularly in the context of finite-dimensional vector spaces, a basis is a set of vectors that can be used to express any vector in the space as a linear combination of those basis vectors.

A **differentiable stack** is a concept arising from the fields of differential geometry, algebraic topology, and category theory, particularly in the context of homotopy theory and advanced mathematical frameworks like derived algebraic geometry. In general, a **stack** is a categorical structure that allows for the systematic handling of "parametrized" objects, facilitating the study of moduli problems in algebraic geometry and related fields.

Discrete differential geometry is a branch of mathematics that studies geometric structures and concepts using discrete analogs rather than continuous ones. It often focuses on the analysis and approximation of geometric properties of surfaces and spaces through polygonal or polyhedral representations, as opposed to smooth manifolds that are typically the focus of classical differential geometry.

A double vector bundle is a mathematical structure that arises in differential geometry and algebraic topology. It generalizes the concept of a vector bundle by considering not just one vector space associated with each point in a manifold, but two layers of vector spaces.

An Equiareal map, also known as an equal-area map, is a type of map projection that maintains the consistency of area proportions across the entire map. This means that regions on the map are represented in the same area ratio as they are on the Earth's surface. As a result, if two areas are equal in size on the map, they will also be equal in size in reality, regardless of their location.

A Kähler-Einstein metric is a special type of Riemannian metric that arises in differential geometry and algebraic geometry. It is associated with Kähler manifolds, which are a class of complex manifolds with a compatible symplectic structure. A Kähler manifold is a complex manifold \( (M, J) \) equipped with a Kähler metric \( g \), which is a Riemannian metric that is both Hermitian and symplectic.

The Equivariant Index Theorem is a significant result in mathematics that generalizes the classical index theorem in the context of equivariant topology, particularly in the presence of group actions. It relates the index of an elliptic differential operator on a manifold equipped with a group action to topological invariants associated with the manifold and the group.

Equivalent latitude is a concept used in atmospheric science and meteorology to describe the latitude corresponding to a particular atmospheric condition or property that is typically associated with a certain latitude in the atmosphere. It is often used in the context of phenomena such as the stratosphere, tropopause, or specific atmospheric trace gases. One common application of equivalent latitude is in the study of the ozone layer and the polar vortex.

The Filling Area Conjecture is a concept from the field of geometric topology, particularly in the study of three-dimensional manifolds. It concerns the relationship between the topological properties of a surface and its geometric properties, specifically focusing on the area of certain types of surfaces. The conjecture originates from the study of isotopy classes of simple curves on surfaces.

A **Finsler manifold** is a generalization of a Riemannian manifold that allows for the length of tangent vectors to be defined in a more flexible way. While Riemannian geometry is based on a positive-definite inner product that varies smoothly from point to point, Finsler geometry introduces a more general function, referred to as the **Finsler metric**, which defines the length of tangent vectors.

In programming, particularly in the context of functional programming and data processing, a "flat map" is a higher-order function that applies a map function to each element of a data structure (like a list or an array), and then flattens the result into a single structure.

A G2-structure is a mathematical concept within the field of differential geometry, particularly in the study of special types of manifolds. More specifically, G2-structures are related to the notion of "exceptional" symmetries and are associated with the G2 group, which is one of the five exceptional Lie groups.

Gaussian curvature is a measure of the intrinsic curvature of a surface at a given point. It is defined as the product of two principal curvatures at that point, which are the maximum and minimum curvatures of the surface in two perpendicular directions.

A geodesic manifold is a type of manifold in differential geometry where the notion of distance and the concept of geodesics, which are the shortest paths between points, can be defined. More specifically, it often refers to a Riemannian manifold equipped with a Riemannian metric, allowing for the computation of distances and angles.

The Haefliger structure, often referred to in the context of differential geometry and topology, is a specific kind of manifold structure that arises in the study of pseudogroups and foliated spaces. It is named after André Haefliger, who contributed significantly to the classification of certain types of smooth structures on manifolds.